Since there is no domain controller on your network there is no centralized security. Running 2000 (I'm guessing Professional) in a peer to peer means each workstation has its own security. This security is not shared with any other machine on the network. When you go to access a resource on another machine on the network the first thing Windows will try to do is use the existing users account information (the person that is currently logged in) to connect. If that username and password do not exist on the machine you are attaching to it will ask you to logon to with a known account(to that machine.)
To resolve this problem you could make all the administrator's password on each machine the same and have the users only log in with thataccount. This could create some problems with people having full access to another persons workstation. The other thing you could do, the one I would do outside of making a Domain Controller, is create user accounts and passwords on each machine forall the users on the network. Then they could attach to shared resources without being prompted for a password while still maintaining some sense of security.
If you want a horror story about a peer to peer scenerio like this then just write some notes over the next few weeks. This setup is going to be a bear to administer and you will be living a horror story with it. My advice: get a domain controller. It doesnt have to do anything but maintain an account database for the users.
